Quantum-critical Pairing with Varying Exponents

We analyze the onset temperature T-p for the pairing in cuprate superconductors at small doping, when tendency towards antiferromagnetism is strong. We consider the model of Moon and Sachdev (MS), which assumes that electron and hole pockets survive in a paramagnetic phase. Within this model, the pairing between fermions is mediated by a gauge boson, whose propagator remains massless in a paramagnet. We relate the MS model to a generic gamma-model of quantum-critical pairing with the pairing kernel lambda(Omega(n)) proportional to 1/Omega(gamma)(n). We show that, over some range of parameters, the MS model is equivalent to gamma=1/3-model (lambda(Omega)proportional to Omega(-1/3)). We find, however, that the parameter range where this analogy works is bounded on both ends. At larger deviations from a magnetic phase, the MS model becomes equivalent to gamma model with varying gamma > 1/3, whose value depends on the distance to a magnetic transition and approaches gamma=1 deep in a paramagnetic phase. Very near the transition, the MS model becomes equivalent to gamma model with varying gamma < 1/3. Right at the magnetic QCP, the MS model is equivalent to the model with lambda(Omega (n))proportional to log Omega(n) , which is the model for color superconductivity. Using this analogy, we verify the formula for T-c derived for color superconductivity.